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UNIT 5 Geometric Relationships MODULE MODULE 9 Applications of Geometry Concepts 7.8.C, 7.9.B, 7.9.C, 7.11.C 10 10 Volume and Surface MODULE MODULE Area 7.8.A, 7.9.A, 7.9.D CAREERS IN MATH Product Design Engineer A product design engineer works to design and develop manufactured products and equipment. A product design engineer uses math to design and modify models, and to calculate costs in producing their designs. © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Juice Images/Alamy If you are interested in a career in product design engineering, you should study these mathematical subjects: • Algebra • Geometry • Trigonometry • Statistics • Calculus Research other careers that require the use of mathematics to design and modify products. Unit 5 Performance Task At the end of the unit, check out how product design engineers use math. Unit 5 277 UNIT 5 Vocabulary Preview 1. NEONGTURC LANSEG 2. LEOTECRAYMPMN SEGLAN 3. RIMEUCEEFNRCC 4. LEARATL ERAA 5. PIECOTMOS GUISEFR 1. Angles that have the same measure. (Lesson 9.1) 2. Two angles whose measures have a sum of 90 degrees. (Lesson 9.1) 3. The distance around a circle. (Lesson 9.4) 4. The sum of the areas of the lateral faces of a prism. (Lesson 10.3) 5. A two-dimensional figure made from two or more geometric figures. (Lesson 9.4) Q: What do you say when you see an empty parrot cage? A: 278 Vocabulary Preview ! © Houghton Mifflin Harcourt Publishing Company Use the puzzle to preview key vocabulary from this unit. Unscramble the circled letters to answer the riddle at the bottom of the page. Applications of Geometry Concepts ? MODULE 9 LESSON 9.1 ESSENTIAL QUESTION Angle Relationships 7.11.C How can you apply geometry concepts to solve real-world problems? LESSON 9.2 Finding Circumference 7.9.B LESSON 9.3 Area of Circles 7.8.C, 7.9.B LESSON 9.4 Area of Composite Figures © Houghton Mifflin Harcourt Publishing Company 7.9.C Real-World Video my.hrw.com my.hrw.com A 16-inch pizza has a diameter of 16 inches. You can use the diameter to find circumference and area of the pizza. You can also determine how much pizza is in one slice of different sizes of pizzas. my.hrw.com Math On the Spot Animated Math Personal Math Trainer Go digital with your write-in student edition, accessible on any device. Scan with your smart phone to jump directly to the online edition, video tutor, and more. Interactively explore key concepts to see how math works. Get immediate feedback and help as you work through practice sets. 279 Are YOU Ready? Personal Math Trainer Complete these exercises to review skills you will need for this chapter. Multiply with Fractions and Decimals EXAMPLE my.hrw.com Online Assessment and Intervention Multiply as you would with whole numbers. Count the total number of decimal places in the two factors. 7.3 × 2.4 292 +146 1 7.5 2 Place the decimal point in the product so that there are the same number of digits after the decimal point. Multiply. 1. 4.16 × 13 _ 2. 6.47 × 0.4 _ 3. 7.05 × 9.4 _ 4. 25.6 × 0.49 __ Area of Squares, Rectangles, and Triangles A = _12 bh EXAMPLE 2.8 cm 7.8 cm Use the formula for area of a triangle. = _12 (7.8) (2.8) Substitute for each variable. = 10.92 cm2 Multiply. Find the area of each figure. 6. square with sides of 3.5 ft 7. rectangle with length 8 _12 in. and width 6 in. 8. triangle with base 12.5 m and height 2.4 m 280 Unit 5 © Houghton Mifflin Harcourt Publishing Company 5. triangle with base 14 in. and height 10 in. Reading Start-Up Visualize Vocabulary Use the ✔ words to complete the graphic. You will put one word in each oval. Line segment with one endpoint on a circle and one at the center of the circle. Distance around a circle. Attributes of Circles Line segment that passes through the center of a circle and has endpoints on the circle. The ratio of circumference to diameter. Understand Vocabulary Complete each sentence, using the preview words. 1. are angles that have the same measure. 2. are two angles whose measures Vocabulary Review Words acute angle (ángulo agudo) ✔ circumference (circunferencia) ✔ diameter (diámetro) obtuse angle (ángulo obtuso) ✔ pi (pi) ✔ radius (radio) right angle (ángulo recto) vertex (vértice) Preview Words adjacent angles (ángulos adyacentes) complementary angles (ángulos complementarios) congruent angles (ángulos congruentes) supplementary angles (ángulos suplementarios) vertical angles (ángulos opuestos por el vértice) © Houghton Mifflin Harcourt Publishing Company have a sum of 90°. Active Reading Four-Corner Fold Before beginning the module, create a four-corner fold to help you organize what you learn. As you study this module, note important ideas, such as vocabulary, properties, and formulas, on the flaps. Use one flap for each lesson in the module. You can use your FoldNote later to study for tests and complete assignments. Module 9 281 MODULE 9 Unpacking the TEKS Understanding the TEKS and the vocabulary terms in the TEKS will help you know exactly what you are expected to learn in this module. What It Means to You Write and solve equations using geometry concepts, including the sum of the angles in a triangle, and angle relationships. You will learn about supplementary, complementary, vertical, and adjacent angles. You will solve simple equations to find the measure of an unknown angle in a figure. Key Vocabulary UNPACKING EXAMPLE 7.11.C supplementary angles (ángulos suplementarios) Two angles whose measures have a sum of 180°. complementary angles (ángulos complementarios) Two angles whose measures add to 90°. vertical angles (ángulos opuestos por el vértice) A pair of opposite congruent angles formed by intersecting lines. 7.9.B Determine the circumference and area of circles. Key Vocabulary area (área) The number of square units needed to cover a given surface. circumference (circunferencia) The distance around a circle. Line n‖ line p. Find the measure of angle 2. Corresponding angles are congruent. 7 m∠1 = 55° p Adjacent angles formed by two intersecting lines are supplementary. 55° 4 n 1 3 2 m∠1 + m∠2 = 180° 55° + m∠2 = 180° Substitute. m∠2 = 180° - 55° = 125° What It Means to You You will use formulas to solve problems involving the area and circumference of circles. UNPACKING EXAMPLE 7.9.B Lily is drawing plans for a circular fountain. The diameter of the fountain is 20 feet. What is the approximate circumference? C = πd C ≈ 3.14 · 20 Substitute. Visit my.hrw.com to see all the unpacked. my.hrw.com 282 Unit 5 6 5 C ≈ 62.8 The circumference of the fountain is about 62.8 feet. © Houghton Mifflin Harcourt Publishing Company 7.11.C LESSON 9.1 Angle Relationships ? Equations, expressions, and relationships—7.11.C Write and solve equations using geometry concepts, including the sum of the angles in a triangle, and angle relationships. ESSENTIAL QUESTION How can you use angle relationships to solve problems? EXPLORE ACTIVITY 7.11.C Measuring Angles It is useful to work with pairs of angles and to understand how pairs of angles relate to each other. Congruent angles are angles that have the same measure. STEP 1 Using a ruler, draw a pair of intersecting lines. Label each angle from 1 to 4. STEP 2 Use a protractor to help you complete the chart. Angle Measure of Angle m∠1 m∠2 m∠3 m∠4 m∠1 + m∠2 m∠2 + m∠3 © Houghton Mifflin Harcourt Publishing Company m∠3 + m∠4 m∠4 + m∠1 Reflect 1. Conjecture Share your results with other students. Make a conjecture about pairs of angles that are opposite each other. 2. Conjecture When two lines intersect to form two angles, what conjecture can you make about the pairs of angles that are next to each other? Lesson 9.1 283 Angle Pairs and One-Step Equations Math On the Spot Vertical angles are the opposite angles formed by two intersecting lines. Vertical angles are congruent because the angles have the same measure. Adjacent angles are pairs of angles that share a vertex and one side but do not overlap. my.hrw.com Complementary angles are two angles whose measures have a sum of 90°. Supplementary angles are two angles whose measures have a sum of 180°. You discovered in the Explore Activity that adjacent angles formed by two intersecting lines are supplementary. EXAMPLE 1 7.11.C Use the diagram. A Name a pair of vertical angles. 50° x ∠AFB and ∠DFE Math Talk Mathematical Processes Are ∠BFD and ∠AFE vertical angles? Why or why not? C B B Name a pair of adjacent angles. ∠AFB and ∠BFD D F A E C Name a pair of supplementary angles. ∠AFB and ∠BFD D Find the measure of ∠AFB. My Notes m∠AFB + m∠BFD = 180° They are supplementary angles. x + 140° = 180° -140° -140° x = 40° m∠BFD = 50° + 90° = 140° Subtract 140° from both sides. The measure of ∠AFB is 40°. Reflect 3. Analyze Relationships What is the relationship between ∠AFB and ∠BFC ? Explain. 4. Draw Conclusions Are ∠AFC and ∠BFC adjacent angles? Why or why not? 284 Unit 5 © Houghton Mifflin Harcourt Publishing Company Use the fact that ∠AFB and ∠BFD in the diagram are supplementary angles to find m∠AFB. YOUR TURN A Use the diagram. 5. Name a pair of supplementary angles. Personal Math Trainer F E 6. Name a pair of vertical angles. B 35° Online Assessment and Intervention my.hrw.com G C D 7. Find the measure of ∠CGD. Angle Pairs and Two-Step Equations Sometimes solving an equation is only the first step in using an angle relationship to solve a problem. Math On the Spot EXAMPL 2 EXAMPLE 7.11.C my.hrw.com A Find the measure of ∠EHF. F E © Houghton Mifflin Harcourt Publishing Company STEP 1 2x 48° H ∠EHF and ∠FHG form a straight line. G Identify the relationship between ∠EHF and ∠FHG. ∠EHF and ∠FHG are supplementary angles. STEP 2 Since angles ∠EHF and ∠FHG form a straight line, the sum of the measures of the angles is 180°. Write and solve an equation to find x. m∠EHF + m∠FHG = 180° 2x + 48° = 180° -48° -48° 2x = 132° x = 66° The sum of the measures of supplementary angles is 180°. Subtract 48° from both sides. Divide both sides by 2. Since m∠EHF = 2x, then m∠EHF = 132°. Lesson 9.1 285 B Find the measure of ∠ZXY. STEP 1 W Identify the relationship between ∠WXZ and ∠ZXY. ∠WXZ and ∠ZXY are complementary angles. Z 35° (4x + 7)° X Y STEP 2 Write and solve an equation to find x. The sum of the measures of m∠WXZ + m∠ZXY = 90° complementary angles is 90°. 35° + (4x + 7)° = 90° Substitute the values. 4x + 42° = 90° Combine like terms. -42° -42° Subtract 42 from both sides. Divide both sides by 4. 4x = 48° x = 12° STEP 3 Find the measure of ∠ZXY. m∠ZXY = (4x + 7)° = (4(12) + 7)° Substitute 12 for x. = 55° Use the Order of Operations. The measure of ∠ZXY is 55°. YOUR TURN 8. Find the measure of ∠JML. J 3x 54° M N x= m∠JML = 3x = 9. Critique Reasoning Cory says that to find m∠JML above you can stop when you get to the solution step 3x = 126°. Explain why this works. Personal Math Trainer Online Assessment and Intervention my.hrw.com 286 Unit 5 © Houghton Mifflin Harcourt Publishing Company L Using Angle Measures in Triangles You learned earlier that the sum of the measures of the angles in any triangle is 180°. You can use this property in many real-world situations. EXAMPL 3 EXAMPLE 7.11.C Math On the Spot my.hrw.com The front of the top story of a house is shaped like an isosceles triangle. The measure of the angle at the top of the triangle is 70°. Find the measure of each of the base angles. STEP 1 Make a sketch. C 70° x x A STEP 2 B Write an equation. m∠A + m∠B + m∠C = 180° © Houghton Mifflin Harcourt Publishing Company • Image Credits: © Christian Beier/CBpictures/Alamy x + STEP 3 x + 70° = 180° The sum of the angle measures in a triangle is 180°. Substitute values. Solve the equation to find x. x + x + 70° = 180° 2x + 70° = 180° - 70° - 70° 2x = 110° 2x ___ 2 110° = ____ 2 x = 55° Combine like terms. Subtract 70° from both sides. Divide both sides of the equation by 2. Each of the base angles measures 55°. YOUR TURN Use the diagram. C 80° 10. Find the value of x. 11. Find the measures of ∠A and ∠B. A x Personal Math Trainer 3x B Online Assessment and Intervention my.hrw.com Lesson 9.1 287 Guided Practice For Exercises 1–2, use the figure. (Example 1) 1. Vocabulary The sum of the measures of ∠UWV and ∠UWZ is 90°, so ∠UWV and ∠UWZ are X V W Y angles. U 2. Vocabulary ∠UWV and ∠VWX share a vertex and one side. They do not overlap, so ∠UWV and ∠VWX are Z angles. For Exercises 3–4, use the figure. B 3. ∠AGB and ∠DGE are angles, so m∠DGE = G . (Example 1) 4. Find the measure of ∠EGF. (Example 2) F + 50° 2x D E m∠CGD + m∠DGE + m∠EGF = 180° + C 30° A = 180° + 2x = 180° 2x = m∠EGF = 2x = B 5. Find the measures of ∠A and ∠B. (Example 3) m∠A + m∠B + m∠C = 180° + = 180 2x + = 180 A 2x = ? ? x= , so m∠A = x + 10 = , so m∠B = x . . ESSENTIAL QUESTION CHECK-IN 6. Suppose that you know that ∠T and ∠S are supplementary, and that m∠T = 3 · (m∠S). How can you find m∠T? 288 Unit 5 40° C © Houghton Mifflin Harcourt Publishing Company + x + 10 Name Class Date 9.1 Independent Practice Personal Math Trainer 7.11.C my.hrw.com For Exercises 7–11, use the figure. T S 41° R P Solve for each indicated angle measure or variable in the figure. I Q U K N 7. Name a pair of adjacent angles. Explain why they are adjacent. Online Assessment and Intervention 84° 4x M G H 12. x 13. m∠KMH Solve for each indicated angle measure or variable in the figure. 8. Name a pair of acute vertical angles. C A 9. Name a pair of supplementary angles. © Houghton Mifflin Harcourt Publishing Company D 62° F 10. Justify Reasoning Find m∠QUR. Justify your answer. B E 14. m∠CBE 15. m∠ABF 16. m∠CBA Solve for each indicated angle measure or variable in the figure. 11. Draw Conclusions Which is greater, m∠TUR or m∠RUQ? Explain. 25° P Q 3x 20° R 17. x 18. m∠Q Lesson 9.1 289 Work Area FOCUS ON HIGHER ORDER THINKING Let △ABC be a right triangle with m∠C = 90°. 19. Critical Thinking An equilateral triangle has three congruent sides and three congruent angles. Can △ABC be an equilateral triangle? Explain your reasoning. 20. Counterexample An isosceles triangle has two congruent sides, and the angles opposite those sides are congruent. River says that right triangle ABC cannot be an isosceles triangle. Give a counterexample to show that his statement is incorrect. 21. Make a Conjecture In a scalene triangle, no two sides have the same length, and no two angles have the same measure. Do you think a right triangle can be a scalene triangle? Explain your reasoning. e. Av 50° Green r Pa k J Ave. 23. Analyze Relationships In triangle XYZ, m∠X = 30°, and all the angles have measures that are whole numbers. Angle Y is an obtuse angle. What is the greatest possible measure that angle Z can have? Explain your answer. 290 Unit 5 © Houghton Mifflin Harcourt Publishing Company 22. Represent Real-World Problems The railroad tracks meet the road as shown. The town will allow a parking lot at angle J if the measure of angle J is greater than 38°. Can a parking lot be built at angle J ? Why or why not? LESSON 9.2 Finding Circumference ? Equations, expressions, and relationships— 7.9.B Determine the circumference … of circles. ESSENTIAL QUESTION How do you find the circumference of a circle? Finding Circumference Using Diameter As you learned earlier, the ratio of the circumference to the diameter _Cd is the same for all circles. This ratio is called π or pi, and you can approximate it as 22 3.14 or as __ 7 . You can use π to find a formula for circumference. For any circle, _Cd = π. Solve the equation for C to give an equation for the Math On the Spot my.hrw.com circumference of a circle in terms of the diameter. C __ =π d The ratio of the circumference to the diameter is π. C __ ×d=π×d d C = πd Multiply both sides by d. Simplify. EXAMPL 1 EXAMPLE 7.9.B Find the circumference of the circle to the nearest hundredth. 22 Use 3.14 or __ 7 for π. STEP 1 Identify the diameter of the circle. 8 in. Math Talk d = 8 in. © Houghton Mifflin Harcourt Publishing Company STEP 2 Mathematical Processes Use the formula. C = πd C = π(8) Substitute 8 for d. C ≈ 3.14(8) Substitute 3.14 for π. C ≈ 25.12 Multiply. Explain why you wouldn’t 22 want to use __ 7 as an approximation for π in this problem. The circumference is about 25.12 inches. Reflect 1. What value of π could you use to estimate the circumference? 2. Checking for Reasonableness How do you know your answer is reasonable? Lesson 9.2 291 YOUR TURN Personal Math Trainer 11 cm 3. Find the circumference of the circle to the nearest hundredth. Online Assessment and Intervention my.hrw.com Finding Circumference Using Radius Since the diameter of a circle is the same as 2 times the radius, you can substitute 2r in the equation for d. Math On the Spot my.hrw.com C = πd C = π(2r) C = 2πr Substitute 2r for d. Use the Commutative Property. The two equivalent formulas for circumference are C = πd or C = 2πr. EXAMPLE 2 7.9.B An irrigation sprinkler waters a circular region with a radius of 14 feet. Find the circumference of the region watered by the 22 sprinkler. Use __ for π. 7 Use the formula. The radius is 14 feet. C = 2π(14) Substitute 14 for r. 22 (14) C ≈ 2(__ 7) 22 for π. Substitute ___ C ≈ 88 Multiply. 7 The circumference of the region watered by the sprinkler is about 88 feet. Reflect 22 4. Analyze Relationships When is it logical to use __ instead of 3.14 7 for π? YOUR TURN Personal Math Trainer Online Assessment and Intervention my.hrw.com 292 Unit 5 5. Find the circumference of the circle. 21 cm © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Corbis C = 2πr 14 ft Using Circumference Given the circumference of a circle, you can use the appropriate circumference formula to find the radius or the diameter of the circle. You can use that information to solve problems. my.hrw.com EXAMPL 3 EXAMPLE 7.9.B A circular pond has a circumference of 628 feet. A model boat is moving directly across the pond, along a radius, at a rate of 5 feet per second. How long does it take the boat to get from the edge of the pond to the center? STEP 1 Math On the Spot My Notes Find the radius of the pond. C = 2πr Use the circumference formula. 628 ≈ 2(3.14)r Substitute for the circumference and for π. 628 6.28r ____ ____ ≈ Divide both sides by 6.28. 6.28 6.28 100 ≈ r C = 628 ft r = ? ft Simplify. The radius is about 100 feet. STEP 2 Find the time it takes the boat to get from the edge of the pond to the center along the radius. Divide the radius of the pond by the speed of the model boat. 100 ÷ 5 = 20 It takes the boat about 20 seconds to get to the center of the pond. Reflect © Houghton Mifflin Harcourt Publishing Company 6. Analyze Relationships Dante checks the answer to Step 1 by multiplying it by 6 and comparing it with the given circumference. Explain why Dante’s estimation method works. Use it to check Step 1. 7. What If? Suppose the model boat were traveling at a rate of 4 feet per second. How long would it take the model boat to get from the edge of the pond to the center? YOUR TURN 8. A circular garden has a circumference of 44 yards. Lars is digging a straight line along a diameter of the garden at a rate of 7 yards per hour. How many hours will it take him to dig across the garden? Personal Math Trainer Online Assessment and Intervention my.hrw.com Lesson 9.2 293 Guided Practice Find the circumference of each circle. (Examples 1 and 2) 1. C = πd 2. C = 2πr C ≈ 2( 9 in. C≈ C≈ 7 cm )( 22 __ 7 ) C≈ inches cm 22 Find the circumference of each circle. Use 3.14 or __ for π. Round to the 7 nearest hundredth, if necessary. (Examples 1 and 2) 3. 4. 5. 4.8 yd 25 m 7.5 in. 6. A round swimming pool has a circumference of 66 feet. Carlos wants to buy a rope to put across the diameter of the pool. The rope costs $0.45 per foot, and Carlos needs 4 feet more than the diameter of the pool. How much will Carlos pay for the rope? (Example 3) Find the diameter. C = πd ≈ 3.14d Find the cost. Carlos needs rope. feet of × $0.45 = 3.14d _____ ≈ _____ 3.14 3.14 Carlos will pay for the rope. Find each missing measurement to the nearest hundredth. Use 3.14 for π. (Examples 1 and 3) 7. r = d= C = π yd ? ? 8. r ≈ d≈ C = 78.8 ft ESSENTIAL QUESTION CHECK-IN 10. Norah knows that the diameter of a circle is 13 meters. How would you tell her to find the circumference? 294 Unit 5 9. r ≈ d ≈ 3.4 in. C= © Houghton Mifflin Harcourt Publishing Company ≈d Name Class Date 9.2 Independent Practice 7.9.B my.hrw.com Find the circumference of each circle. Use 3.14 22 or __ for π. Round to the nearest hundredth, 7 if necessary. 11. 5.9 ft 12. Personal Math Trainer 56 cm Online Assessment and Intervention 18. Multistep Randy’s circular garden has a radius of 1.5 feet. He wants to enclose the garden with edging that costs $0.75 per foot. About how much will the edging cost? Explain. 19. Represent Real-World Problems The Ferris wheel shown makes 12 revolutions per ride. How far would someone travel during one ride? 13. 35 in. diameter 63 feet © Houghton Mifflin Harcourt Publishing Company 14. In Exercises 11–13, for which problems did 22 you use __ for π? Explain your choice. 7 20. The diameter of a bicycle wheel is 2 feet. About how many revolutions does the wheel make to travel 2 kilometers? Explain. Hint: 1 km = 3,280 ft 15. A circular fountain has a radius of 9.4 feet. Find its diameter and circumference to the nearest hundredth. 16. Find the radius and circumference of a CD with a diameter of 4.75 inches. 17. A dartboard has a diameter of 18 inches. What is its radius and circumference? 21. Multistep A map of a public park shows a circular pond. There is a bridge along a diameter of the pond that is 0.25 mi long. You walk across the bridge, while your friend walks halfway around the pond to meet you at the other side of the bridge. How much farther does your friend walk? Lesson 9.2 295 22. Architecture The Capitol Rotunda connects the House and the Senate sides of the U.S. Capitol. Complete the table. Round your answers to the nearest foot. Capitol Rotunda Dimensions Height Circumference 180 ft 301.5 ft Radius Diameter FOCUS ON HIGHER ORDER THINKING Work Area 23. Multistep A museum groundskeeper is creating a semicircular statuary garden with a diameter of 30 feet. There will be a fence around the garden. The fencing costs $9.25 per linear foot. About how much will the fencing cost altogether? 24. Critical Thinking Sam is placing rope lights around the edge of a circular patio with a diameter of 18 feet. The lights come in lengths of 54 inches. How many strands of lights does he need to surround the patio edge? 25. Represent Real-World Problems A circular path 2 feet wide has an inner diameter of 150 feet. How much farther is it around the outer edge of the path than around the inner edge? 27. Persevere in Problem Solving Consider two circular swimming pools. Pool A has a radius of 12 feet, and Pool B has a diameter of 7.5 meters. Which pool has a greater circumference? How much greater? Justify your answers. 296 Unit 5 © Houghton Mifflin Harcourt Publishing Company 26. Critique Reasoning A gear on a bicycle has the shape of a circle. One gear has a diameter of 4 inches, and a smaller one has a diameter of 2 inches Justin says that the circumference of the larger gear is 2 inches more than the circumference of the smaller gear. Do you agree? Explain your answer. LESSON 9.3 Area of Circles ? Equations, expressions, and relationships—7.8.C Use models to determine the approximate formulas for the … area of a circle and connect the models to the actual formulas. Also 7.9.B ESSENTIAL QUESTION How do you find the area of a circle? EXPLORE ACTIVITY 1 7.8.C Exploring Area of Circles You can use what you know about circles and π to help find the formula for the area of a circle. STEP 1 Use a compass to draw a circle and cut it out. STEP 2 Fold the circle three times as shown to get equal wedges. STEP 3 Unfold and shade one-half of the circle. STEP 4 Cut out the wedges, and fit the pieces together to form a figure that looks like a parallelogram. The base and height of the parallelogram relate to the parts of the circle. Radius base b = _____ the circumference of the circle, or © Houghton Mifflin Harcourt Publishing Company Half the circumference height h = the of the circle, or To find the area of a parallelogram, the equation is A = To find the area of the circle, substitute for b and h in the area formula. Reflect 1. How can you make the wedges look more like a parallelogram? . A = bh A = h Substitute πr for b. A = πr Substitute r for h. A=π r · r = r2 Lesson 9.3 297 Finding the Area of a Circle Area of a Circle Math On the Spot The area of a circle is equal to π times the radius squared. my.hrw.com r A = πr 2 Remember that area is given in square units. EXAMPLE 1 7.9.B Math Talk Mathematical Processes If the radius increases by 1 centimeter, how does the area of the top of the biscuit change? A = πr2 Use the formula. A = π(4)2 Substitute. Use 4 for r. A ≈ 3.14 × 42 Substitute. Use 3.14 for π. A ≈ 3.14 × 16 Evaluate the power. A ≈ 50.24 Multiply. The area of the biscuit is about 50.24 cm2. Reflect 2. Compare finding the area of a circle when given the radius with finding the area when given the diameter. 3. Personal Math Trainer Online Assessment and Intervention my.hrw.com 298 Unit 5 Why do you evaluate the power in the equation before multiplying? YOUR TURN 4. A circular pool has a radius of 10 feet. What is the area of the pool? Use 3.14 for π. ª)PVHIUPO.JGGMJO)BSDPVSU1VCMJTIJOH$PNQBOZt*NBHF$SFEJUTZigzag Mountain "SU4IVUUFSTUPDL A biscuit recipe calls for the dough to be rolled out and circles to be cut from the dough. The biscuit cutter has a radius of 4 cm. Find the area of the biscuit once it is cut. Use 3.14 for π. EXPLORE ACTIVITY 2 7.9.B Finding the Relationship between Circumference and Area r You can use what you know about circumference and area of circles to find a relationship between them. Find the relationship between the circumference and area of a circle. Start with a circle that has radius r. r = _____ Solve the equation C = 2πr for r. ( ) ( ) 2 Substitute your expression for r in the formula for area of a circle. A = π _____ Remember: Because the exponent is outside the parentheses, you must apply it to the numerator and to each factor of the denominator. 2 A = π _____________ 2 2 · Square the term in the parentheses. 2 · A = ____________2 · © Houghton Mifflin Harcourt Publishing Company Evaluate the power. 2 Simplify. A = ___________ · Solve for C2. C2 = 4 The circumference of the circle squared is equal to . Lesson 9.3 299 EXPLORE ACTIVITY 2 (cont’d) Reflect 5. Does this formula work for a circle with a radius of 3 inches? Show your work. Guided Practice Find the area of each circle. Round to the nearest tenth if necessary. Use 3.14 for π. (Explore Activity 1) 1. 2. 14 m 3. 20 yd 12 mm 4. A clock face has a radius of 8 inches. What is the area of the clock face? Round your answer to the nearest hundredth. 5. A DVD has a diameter of 12 centimeters. What is the area of the DVD? Round your answer to the nearest hundredth. 6. A company makes steel lids that have a diameter of 13 inches. What is the area of each lid? Round your answer to the nearest hundredth. Find the area of each circle. Give your answers in terms of π. (Explore Activity 2) 7. C = 4π 8. C = 12π A= A= π 9. C = __ 2 A= 10. A circular pen has an area of 64π square yards. What is the circumference of the pen? Give your answer in terms of π. (Explore Activity 2) ? ? ESSENTIAL QUESTION CHECK-IN 11. What is the formula for the area A of a circle in terms of the radius r? 300 Unit 5 © Houghton Mifflin Harcourt Publishing Company Solve. Use 3.14 for π. (Example 1) Name Class Date 9.3 Independent Practice 7.8.C, 7.8.B Personal Math Trainer my.hrw.com 12. The most popular pizza at Pavone’s Pizza is the 10-inch personal pizza with one topping. What is the area of a pizza with a diameter of 10 inches? Round your answer to the nearest hundredth. 13. A hubcap has a radius of 16 centimeters. What is the area of the hubcap? Round your answer to the nearest hundredth. 16 cm Online Assessment and Intervention 17. Multistep The sides of a square field are 12 meters. A sprinkler in the center of the field sprays a circular area with a diameter that corresponds to a side of the field. How much of the field is not reached by the sprinkler? Round your answer to the nearest hundredth. 18. Justify Reasoning A small silver dollar pancake served at a restaurant has a circumference of 2π inches. A regular pancake has a circumference of 4π inches. Is the area of the regular pancake twice the area of the silver dollar pancake? Explain. © Houghton Mifflin Harcourt Publishing Company 14. A stained glass window is shaped like a semicircle. The bottom edge of the window is 36 inches long. What is the area of the stained glass window? Round your answer to the nearest hundredth. 15. Analyze Relationships The point (3, 0) lies on a circle with the center at the origin. What is the area of the circle to the nearest hundredth? 16. Multistep A radio station broadcasts a signal over an area with a radius of 50 miles. The station can relay the signal and broadcast over an area with a radius of 75 miles. How much greater is the area of the broadcast region when the signal is relayed? Round your answer to the nearest square mile. 19. Analyze Relationships A bakery offers a small circular cake with a diameter of 8 inches. It also offers a large circular cake with a diameter of 24 inches. Does the top of the large cake have three times the area of that of the small cake? If not, how much greater is its area? Explain. Lesson 9.3 301 2 C 20. Communicate Mathematical Ideas You can use the formula A = __ 4π to find the area of a circle given the circumference. Describe another way to find the area of a circle when given the circumference. 21. Draw Conclusions Mark wants to order a pizza. Which is the better deal? Explain. Donnie’s Pizza Palace Diameter (in.) 12 18 Cost ($) 10 20 22. Multistep A bear was seen near a campground. Searchers were dispatched to the region to find the bear. a. Assume the bear can walk in any direction at a rate of 2 miles per hour. Suppose the bear was last seen 4 hours ago. How large an area must the searchers cover? Use 3.14 for π. Round your answer to the nearest square mile. b. What If? How much additional area would the searchers have to cover if the bear were last seen 5 hours ago? 23. Analyze Relationships Two circles have the same radius. Is the combined area of the two circles the same as the area of a circle with twice the radius? Explain. 24. Look for a Pattern How does the area of a circle change if the radius is multiplied by a factor of n, where n is a whole number? 25. Represent Real World Problems The bull’s-eye on a target has a diameter of 3 inches. The whole target has a diameter of 15 inches. What part of the whole target is the bull’s-eye? Explain. 302 Unit 5 Work Area © Houghton Mifflin Harcourt Publishing Company FOCUS ON HIGHER ORDER THINKING LESSON 9.4 ? Area of Composite Figures Equations, expressions, and relationships— 7.9.C Determine the area of composite figures containing combinations of rectangles, squares, parallelograms, trapezoids, triangles, semicircles, and quarter circles. ESSENTIAL QUESTION How do you find the area of composite figures? 7.9.C EXPLORE ACTIVITY Exploring Areas of Composite Figures Aaron was plotting the shape of his garden on grid paper. While it was an irregular shape, it was perfect for his yard. Each square on the grid represents 1 square meter. A Describe one way you can find the area of this garden. B The area of the garden is square meters. © Houghton Mifflin Harcourt Publishing Company C Compare your results with other students. What other methods were used to find the area? D How does the area you found compare with the area found using different methods? Reflect 1. Use dotted lines to show two different ways Aaron’s garden could be divided up into simple geometric figures. Lesson 9.4 303 Finding the Area of a Composite Figure Math On the Spot my.hrw.com A composite figure is made up of simple geometric shapes. To find the area of a composite figure or other irregular-shaped figure, divide it into simple, nonoverlapping figures. Find the area of each simpler figure, and then add the areas together to find the total area of the composite figure. Use the chart below to review some common area formulas. Shape Area Formula triangle A = _12 bh square A = s2 rectangle A = ℓw parallelogram A = bh trapezoid A = _12 h(b1 + b2) EXAMPLE 1 7.9.C Find the area of the figure. Separate the figure into smaller, familiar figures: a parallelogram 3 cm and a trapezoid. STEP 1 my.hrw.com Area of the Parallelogram 10 cm 1.5 cm 3 cm 1.5 cm 7 cm 2 cm height = 1.5 cm height = 1.5 cm Use the formula. A = 10 · 1.5 The top base of the trapezoid is 10 cm 1 _ since it is the same A = 2 h(b1 + b2) length as the base of A = _12 (1.5)(7 + 10) the parallelogram. A = 15 A = _12 (1.5)(17) = 12.75 The area of the parallelogram is 15 cm2. The area of the trapezoid is 12.75 cm2. base2 = 10 cm Use the formula. Add the areas to find the total area. The area of the figure is 27.75 cm2. Unit 5 4 cm base1 = 7 cm A = 15 + 12.75 = 27.75 cm2 304 Area of the Trapezoid base = 10 cm A = bh STEP 3 2 cm 1.5 cm 7 cm 4 cm Find the area of each shape. STEP 2 10 cm 1.5 cm © Houghton Mifflin Harcourt Publishing Company Animated Math YOUR TURN Find the area of each figure. Use 3.14 for π. 2. Personal Math Trainer 3. 2 ft Online Assessment and Intervention 8 ft my.hrw.com 3 ft 4 ft 10 m 3 ft 8 ft 3 ft 10 m Using Area to Solve Problems EXAMPL 2 EXAMPLE 7.9.C Math On the Spot © Houghton Mifflin Harcourt Publishing Company A banquet room is being carpeted. A floor plan of the room is shown at right. Each unit represents 1 yard. The carpet costs $23.50 per square yard. How much will it cost to carpet the room? my.hrw.com STEP 1 Separate the composite figure into simpler shapes as shown by the dashed lines: a parallelogram, a rectangle, and a triangle. STEP 2 Find the area of the simpler figures. Count units to find the dimensions. STEP 3 Parallelogram Rectangle Triangle A = bh A = ℓw A = _12 bh A = 4 · 2 A = 6 · 4 A = _12 (1)(2) A = 8 yd2 A = 24 yd2 A = 1 yd2 Math Talk Mathematical Processes Describe how you can estimate the cost to carpet the room. Find the area of the composite figure. A = 8 + 24 + 1 = 33 square yards STEP 4 Calculate the cost to carpet the room. Area · Cost per yard = Total cost 33 · $23.50 = $775.50 The cost to carpet the banquet room is $775.50. Lesson 9.4 305 YOUR TURN Personal Math Trainer Online Assessment and Intervention my.hrw.com 4. A window is being replaced with tinted glass. The plan at the right shows the design of the window. Each unit length represents 1 foot. The glass costs $28 per square foot. How much will it cost to replace the glass? Use 3.14 for π. Guided Practice 1. A tile installer plots an irregular shape on grid paper. Each square on the grid represents 1 square centimeter. What is the area of the irregular shape? (Explore Activity, Example 2) STEP 1 Separate the figure into a triangle, a and a parallelogram. STEP 2 Find the area of each figure. triangle: STEP 3 cm2; rectangle: , cm2; parallelogram: + Find the area of the composite figure: The area of the irregular shape is + cm2 = cm2 cm2. 12 cm 2. Show two different ways to divide the composite figure. Find the area both ways. Show your work below. (Example 1) 9 cm 8 cm 18 cm 20 cm 3. Sal is tiling his entryway. The floor plan is drawn on a unit grid. Each unit length represents 1 foot. Tile costs $2.25 per square foot. How much will Sal pay to tile his entryway? (Example 2) ? ? ESSENTIAL QUESTION CHECK-IN 4. What is the first step in finding the area of a composite figure? 306 Unit 5 © Houghton Mifflin Harcourt Publishing Company 9 cm Name Class Date 9.4 Independent Practice Personal Math Trainer 7.9.C my.hrw.com 5. A banner is made of a square and a semicircle. The square has side lengths of 26 inches. One side of the square is also the diameter of the semicircle. What is the total area of the banner? Use 3.14 for π. Online Assessment and Intervention 8. A field is shaped like the figure shown. What is the area of the field? Use 3.14 for π. 8m 8m 6. Multistep Erin wants to carpet the floor of her closet. A floor plan of the closet is shown. 10 ft 4 ft 3 ft 6 ft a. How much carpet does Erin need? © Houghton Mifflin Harcourt Publishing Company b. The carpet Erin has chosen costs $2.50 per square foot. How much will it cost her to carpet the floor? 7. Multiple Representations Hexagon ABCDEF has vertices A(-2, 4), B(0, 4), C(2, 1), D(5, 1), E(5, -2), and F(-2, -2). Sketch the figure on a coordinate plane. What is the area of the hexagon? 8m 9. A bookmark is shaped like a rectangle with a semicircle attached at both ends. The rectangle is 12 cm long and 4 cm wide. The diameter of each semicircle is the width of the rectangle. What is the area of the bookmark? Use 3.14 for π. 10. Multistep Alex is making 12 pennants for the school fair. The pattern he is using to make the pennants is shown in the figure. The fabric for the pennants costs $1.25 per square foot. How much will it cost Alex to make 12 pennants? 1 ft 3 ft 1 ft y 2 -4 -2 O 2 4 11. Reasoning A composite figure is formed by combining a square and a triangle. Its total area is 32.5 ft2. The area of the triangle is 7.5 ft2. What is the length of each side of the square? Explain. -4 Lesson 9.4 307 Work Area FOCUS ON HIGHER ORDER THINKING 12. Represent Real-World Problems Christina plotted the shape of her garden on graph paper. She estimates that she will get about 15 carrots from each square unit. She plans to use the entire garden for carrots. About how many carrots can she expect to grow? Explain. 13. Analyze Relationships The figure shown is made up of a triangle and a square. The perimeter of the figure is 56 inches. What is the area of the figure? Explain. 10 in. 28 in. 15. Persevere in Problem Solving The design for the palladium window shown includes a semicircular shape at the top. The bottom is formed by squares of equal size. A shade for the window will extend 4 inches beyond the perimeter of the window, shown by the dashed line around the window. Each square in the window has an area of 100 in2. a. What is the area of the window? Use 3.14 for π. b. What is the area of the shade? Round your answer to the nearest whole number. 308 Unit 5 15 in. © Houghton Mifflin Harcourt Publishing Company 14. Critical Thinking The pattern for a scarf is shown at right. What is the area of the scarf? Use 3.14 for π. 10 in. 8 in. MODULE QUIZ Ready Personal Math Trainer 9.1 Angle Relationships Online Assessment and Intervention my.hrw.com Use the diagram to name a pair of each type of angle. A 1. Supplementary angles 2. Complementary angles B E C 3. Vertical angles D F 9.2, 9.3 Finding Circumference and Area of Circles Find the circumference and area of each circle. Use 3.14 for π. 36 cm 7m 4. circumference ≈ 6. circumference ≈ 5. area ≈ 7. area ≈ 9.4 Area of Composite Figures Find the area of each figure. Use 3.14 for π. 8. 10 m 9. 4.5 cm © Houghton Mifflin Harcourt Publishing Company 14 m 5.5 cm 20 cm ESSENTIAL QUESTION 10. How can you use geometric formulas in real-world situations? Module 9 309 Personal Math Trainer MODULE 9 MIXED REVIEW Texas Test Prep Selected Response Use the diagram for Exercises 1–3. my.hrw.com 5. What is the circumference of the circle? Use 3.14 for π. B A Online Assessment and Intervention 11 m C 72° F A 34.54 m E D 1. What is the measure of ∠BFC? A 18 ° C 108 ° B 72 ° D 144 ° B 69.08 m C 379.94 m D 1,519.76 m 6. What is the area of the circle? Use 3.14 for π. 15 m 2. Which describes the relationship between ∠BFA and ∠CFD? A adjacent angles B complementary angles supplementary angles D vertical angles C B 47.1 m2 D 706.5 m2 176.625 m2 Gridded Response 3. Which information would allow you to identify ∠BFA and ∠AFE as complementary angles? 7. Find the area in square meters of the figure below? Use 3.14 for π. 6m A m∠AFE = 108 ° B ∠DFE is a right angle. C D ∠BFA and ∠BFC are adjacent angles. 4. David pays $7 per day to park his car. He uses a debit card each time. By what amount does his bank account change due to parking charges over a 40-day period? 310 6m ∠BFA and ∠BFC are supplementary angles. A -$280 C B -$47 D $280 Unit 5 $47 . 0 0 0 0 0 0 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 7 7 7 7 7 7 8 8 8 8 8 8 9 9 9 9 9 9 © Houghton Mifflin Harcourt Publishing Company C A 23.55 m2 Volume and Surface Area ? MODULE 10 LESSON 10.1 ESSENTIAL QUESTION Volume of Rectangular Prisms and Pyramids How can you use volume and surface area to solve real-world problems? 7.8.A, 7.9.A LESSON 10.2 Volume of Triangular Prisms and Pyramids 7.8.A, 7.9.A LESSON 10.3 Lateral and Total Surface Area 7.9.A, 7.9.D © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©nagelestock. com/Alamy Images Real-World Video my.hrw.com my.hrw.com There are famous pyramids around the world. The glass and metal square pyramid in the courtyard of the Louvre Palace in Paris is the main entrance to the Louvre Museum. my.hrw.com Math On the Spot Animated Math Personal Math Trainer Go digital with your write-in student edition, accessible on any device. Scan with your smart phone to jump directly to the online edition, video tutor, and more. Interactively explore key concepts to see how math works. Get immediate feedback and help as you work through practice sets. 311 Are YOU Ready? Personal Math Trainer Complete these exercises to review skills you will need for this chapter. Write a Mixed Number as an Improper Fraction EXAMPLE 3 _58 = 1 + 1 + 1 + _58 my.hrw.com Online Assessment and Intervention Write the whole number as a sum of ones. = _88 + _88 + _88 + _58 Use the denominator of the fraction to write equivalent fractions for the ones. 29 = __ 8 Add the numerators. Write each mixed number as an improper fraction. 2. 2 _34 9 1. 1 __ 10 3. 4 _16 4. 7 _29 Use Repeated Multiplication EXAMPLE 3×3× 3 ↓ ↓ 9 × 3 ↓ 27 Multiply the first two factors. Multiply the result by the next factor. Continue until there are no more factors to multiply. Find each product. 5×5×5 6. 7. 5.1 × 5.1 × 5.1 8. 4×4×4×4 Multiply Fractions EXAMPLE 1 1 ∕ 3∕ 5 __ 3 _ × 10 = _56 ×__ ∕ 6 ∕ 10 2 Divide by the common factors. 2 = _14 Simplify. Multiply. Write each product in simplest form. 9. _12 × _13 3 13. _47 × __ 16 312 Unit 5 10. _34 × _56 15 9 11. __ × __ 16 10 5 12. _25 × __ 12 5 3 14. __ × __ 12 20 15. _38 × _35 7 16. _59 × __ 10 © Houghton Mifflin Harcourt Publishing Company 5. 8 × 8 × 8 Reading Start-Up Visualize Vocabulary Use the ✔ words to complete the graphic. You will put one word in each oval. Volume of a Prism V = Bh Vocabulary Review Words ✔ base (base) face (cara) ✔ height (altura) prism (prisma) rectangular prism (prisma rectangular) right triangle (triángulo rectángulo) triangular prism (prisma triangular) ✔ volume (volumen) Preview Words The capacity of a three-dimensional shape. The area of the bottom of a three-dimensional shape. Distance from the bottom to the top of a shape measured perpendicular to the base. lateral area (área lateral) lateral faces (cara lateral) net (plantlla) pyramid (pirámide) surface area (área total) Understand Vocabulary © Houghton Mifflin Harcourt Publishing Company Match the term on the left to the correct expression on the right. 1. lateral faces A. Parallelograms that form the sides of a prism and connect the bases. 2. surface area B. The sum of the areas of all lateral faces in a prism. 3. lateral area C. The sum of all of the areas of all of the faces of a prism. Active Reading Booklet Before beginning the module, create a booklet to help you learn the concepts in this module. Write the main idea of each lesson on one page of the booklet. As you study each lesson, write important details that support the main idea, such as vocabulary and formulas. Refer to your finished booklet as you work on assignments and study for tests. Module 10 313 MODULE 10 Unpacking the TEKS Understanding the TEKS and the vocabulary terms in the TEKS will help you know exactly what you are expected to learn in this module. 7.9.A Solve problems involving the volume of rectangular prisms, triangular prisms, rectangular pyramids, and triangular pyramids. Key Vocabulary volume (volumen) The number of cubic units needed to fill a given space. What It Means to You You will use edge lengths to find the volume of prisms and pyramids. UNPACKING EXAMPLE 7.9.A Maurice sets up a tent that has the shape of a rectangular pyramid. The base is 7 feet by 6 feet, and the height is 5 feet. What is the volume of the tent? B = base area; Vpyramid = _13 Bh h = height 1 _ Vpyramid = 3 (7 · 6) · 5 = _31 · 210 = 70 7.9.D Solve problems involving the lateral and total surface area of a rectangular prism, rectangular pyramid, triangular prism, and triangular pyramid by determining the area of the shape’s net. Key Vocabulary net (plantilla) An arrangement of twodimensional figures that can be folded to form a polyhedron. surface area (área total) The sum of the areas of the faces, or surfaces, of a threedimensional figure. Visit my.hrw.com to see all the unpacked. my.hrw.com 314 Unit 5 What It Means to You You will use nets to find the surface area of three-dimensional figures in real-world and mathematical problems. UNPACKING EXAMPLE 7.9.D Julie is wrapping a present for a friend. 5 in. Find the surface area of the box. Draw a net to help you see each face of the prism. 5 in. 11 in. 5 in. A 11 in. 21 in. B C D F E 5 in. 21 in. 11 in. Use the formula A = lw to find the area of each face. A: A = 11 × 5 = 55 B: A = 21 × 11 = 231 C: A = 21 × 5 = 105 D: A = 21 × 11 = 231 E: A = 21 × 5 = 105 F: A = 11 × 5 = 55 S = 55 + 231 + 105 + 231 + 105 + 55 = 782 The surface area is 782 in2. © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Morgan Lane Photography/Alamy Images The volume of the tent is 70 ft3. LESSON 10.1 ? Volume of Rectangular Prisms and Pyramids ESSENTIAL QUESTION Equations, expressions, and relationships—7.8.A Model the relationship between the volume of a rectangular prism and a rectangular pyramid having both congruent bases and heights and connect that relationship to the formulas. Also 7.9.A How do you find the volume of a rectangular prism and a rectangular pyramid? Finding the Volume of a Rectangular Prism Remember that the volume of a rectangular prism is given by the formula V = ℓ × w × h, or V = ℓwh. Math On the Spot The base of a rectangular prism is a rectangle with length ℓ and width w, so the area of the base B is equal to ℓw. The volume formula can also be written as V = Bh. In fact, the volume of any prism is the product of the base B and the height h. Remember that volume is given in cubic units. my.hrw.com Volume of a Prism The volume V of a prism is the area of its base B times its height h. © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Photodisc/ Getty Images V = Bh EXAMPL 1 EXAMPLE 7.9.A Find the volume of the rectangular prism. STEP 1 7 mm Find the area of the base. B = ℓw Use the formula. B = 12 × 15 Substitute for ℓ and w. 15 mm 12 mm Mathematical Processes Does it matter which dimension of a rectangle you consider its height? B = 180 mm2 STEP 2 Math Talk Find the volume. V = Bh Use the formula. V = 180 × 7 Substitute for B and h. V = 1,260 mm3 The volume of the rectangular prism is 1,260 cubic millimeters. Lesson 10.1 315 Reflect 1. What If? If you know the volume V and the height h of a prism, how would you find the area of the base B? YOUR TURN Personal Math Trainer 2. Use the formula V = Bh to find the volume of a gift box that is 3.5 inches high, 7 inches long, and 6 inches wide. Online Assessment and Intervention my.hrw.com EXPLORE ACTIVITY 1 7.9.A Nets A net is a two-dimensional pattern of shapes that can be folded into a three-dimensional figure. The shapes in the net become the faces of the three-dimensional figure. STEP 1 Copy Net A and Net B on graph paper, and Net A cut them out along the blue lines. Net B One of these nets can be folded along the black lines to make a cube. Which net will STEP 2 See if you can find another net that can be folded into a cube. Draw a net that you think will make a cube on your graph paper, and then cut it out. Can you fold it into a cube? Sketch your net below. STEP 3 316 Unit 5 Compare your results with several of your classmates. How many different nets for a cube did you and your classmates find? © Houghton Mifflin Harcourt Publishing Company not make a cube? Reflect 3. What shapes will appear in a net for a rectangular prism that is not a cube? How many of these shapes will there be? How do you know that each net cannot be folded into a cube without actually cutting and folding it? 4. 5. © Houghton Mifflin Harcourt Publishing Company 6. Make a Conjecture If you draw a net for a cylinder, such as a soup can, how many two-dimensional geometric shapes would this net have? Name the shapes in the net for a cylinder. Exploring the Volume of a Rectangular Pyramid A pyramid is a three-dimensional shape whose base is a polygon and whose other faces are all triangles. Like a prism, a pyramid is named by the shape of its base. Rectangular Pyramid Triangular Pyramid The faces of a pyramid that are not the base have a common vertex, called the vertex of the pyramid. The perpendicular distance from the vertex to the base is the height of the pyramid. Pentagonal Pyramid Height Base Lesson 10.1 317 EXPLORE ACTIVITY 2 7.8.A In this activity, you will compare the volumes of a pyramid and a prism with congruent bases and equal heights. Remember that congruent figures have the same shape and size. STEP 1 Make three-dimensional models. Make larger versions of the nets shown. Make sure the bases and heights in each net are the same size. Fold each net, and tape it together to form a prism or a pyramid. STEP 2 Fill the pyramid with beans. Make sure that the beans are level with the opening of the pyramid. Then pour the beans into the prism. Repeat until the prism is full. How many times did you fill the prism from the pyramid? STEP 3 Write a fraction that compares the volume of the pyramid to the volume of the prism. volume of pyramid _____ ________________ = volume of prism Math Talk Mathematical Processes Describe ways in which a prism and a pyramid are different. Reflect 8. Communicate Mathematical Ideas The prism and the pyramid in this activity have congruent bases and equal heights. Are they congruent three-dimensional shapes? Explain. YOUR TURN Personal Math Trainer Online Assessment and Intervention my.hrw.com 318 Unit 5 3 9. The volume of a rectangular prism is 4_12 in . What is the volume of a rectangular pyramid with a congruent base and the same height? Explain your reasoning. © Houghton Mifflin Harcourt Publishing Company 7. Draw Conclusions A rectangular pyramid has a base area of B and a height of h. What is a formula for the volume of the pyramid? Justify your reasoning. Solving Volume Problems You can use the formulas for the volume of a rectangular prism and the volume of a rectangular pyramid to solve problems. Math On the Spot Volume of a Rectangular Pyramid my.hrw.com The volume V of a pyramid is one-third the area of its base B times its height h. V = _13 Bh EXAMPL 2 EXAMPLE 7.9.A A Kyle needs to build a crate in the shape of a rectangular prism. The crate must have a volume of 38_12 cubic feet, and a base area of 15_25 square feet. Find the height of the crate. V = Bh 38_12 = 15_25 · h Use the formula. Substitute for V and B. 77 __ __ = 77 h 5 2 Change the mixed numbers to fractions. 5 __ 5 77 __ · 77 = __ h · __ 77 2 5 77 77 , multiply both To divide both sides by ___ 5 sides by the reciprocal. 5 _ =h 2 © Houghton Mifflin Harcourt Publishing Company The height of the crate must be _52, or 2_12, feet. B A glass paperweight in the shape of a rectangular pyramid has a base that is 4 inches by 3 inches and a height of 5 inches. Find the volume of the paperweight. V = _13 Bh Use the formula. V = _13 · 12 · 5 Think: B = ℓw = 4 · 3 = 12 V = 20 The paperweight has a volume of 20 cubic inches. YOUR TURN 10. A rectangular prism has a volume of 160 cubic centimeters and a height of 4 centimeters. What is the area of its base? 11. A square pyramid has a base edge of 5.5 yards and a height of 3.25 yards. Find the volume of the pyramid to the nearest tenth. Personal Math Trainer Online Assessment and Intervention my.hrw.com Lesson 10.1 319 Guided Practice 1. Find the volume of the rectangular prism. (Example 1) V = Bh ( V= V= × _____ ) ( ) 9 ft 6 ft 1 ft 4 2 ft3 Identify the three-dimensional shape that can be formed from each net. (Explore Activity 1 and Explore Activity 2) 3. 5. The volume of a rectangular prism is 161.2 m3. The prism has a base that is 5.2 m by 3.1 m. Find the height of the prism. (Example 2) 4. V = Bh = ( = ) × ×h h =h The height of the prism is ? ? . ESSENTIAL QUESTION CHECK-IN 6. Explain how to use models to show the relationship between the volume of a rectangular prism and a rectangular pyramid with congruent bases and heights. 320 Unit 5 © Houghton Mifflin Harcourt Publishing Company 2. Name Class Date 10.1 Independent Practice Personal Math Trainer 7.8.A, 7.9.A 8. Explain the Error A student found the volume of a rectangular pyramid with a base area of 92 square meters and a height of 54 meters to be 4,968 cubic meters. Explain and correct the error. my.hrw.com 13. A storage chest has the shape of a rectangular prism with the dimensions shown. The volume of the storage chest is 18,432 cubic inches. What is its height? 16 in. 9. A block of marble is in the shape of a rectangular prism. The block is 3 feet long, 2 feet wide, and 18 inches high. What is the volume of the block? 10. Multistep Curtis builds a doghouse with base shaped like a cube and a roof shaped like a pyramid. The cube has an edge length of 3_12 feet. The height of the pyramid is 5 feet. Find the volume of the doghouse rounded to the nearest tenth. Online Assessment and Intervention 48 in. 14. Draw Conclusions A shipping company ships certain boxes at a special rate. The boxes must not have a volume greater than 2,500 cm3. Can the box shown be shipped at the special rate? Explain. 20 cm 10 cm © Houghton Mifflin Harcourt Publishing Company 12 cm 15. Communicate Mathematical Ideas Is the figure shown a prism or a pyramid? Justify your answer. 11. Miguel has an aquarium in the shape of a rectangular prism. The base is 30.25 inches long and 12.5 inches wide. The aquarium is 12.75 inches high. What is the volume of the aquarium to the nearest cubic inch? 11 in. 9 in. 4 in. 8 in. 12. After a snowfall, Sheree built a snow pyramid. The pyramid had a square base with side lengths of 32 inches and a height of 28 inches. What was the volume of the pyramid to the nearest cubic inch? Lesson 10.1 321 16. A cereal box can hold 144 cubic inches of cereal. Suppose the box is 8 inches long and 1.5 inches wide. How tall is the box? 17. A shed in the shape of a rectangular prism has a volume of 1,080 cubic feet. The height of the shed is 8 feet, and the width of its base is 9 feet. What is the length of the shed? FOCUS ON HIGHER ORDER THINKING Work Area 18. Draw Conclusions Sue has a plastic paperweight shaped like a rectangular pyramid. The volume is 120 cubic inches, the height is 6 inches, and the length is 10 inches. She has a gift box that is a rectangular prism with a base that is 6 inches by 10 inches. How tall must the box be for it to hold the pyramid? 19. Represent Real-World Problems A public swimming pool is in the shape of a rectangular prism. The pool is 20 meters long and 16 meters wide. The pool is filled to a depth of 1.75 meters. a. Find the volume of water in the pool. b. A cubic meter of water has a mass of 1,000 kilograms. Find the mass of the water in the pool. 21. Persevere in Problem Solving A small solid pyramid was installed on top of the Washington Monument in 1884. The square base of the pyramid is 13.9 centimeters on a side, and the height of the pyramid is 22.6 centimeters. The pyramid has a mass of 2.85 kilograms. a. Find the volume of the pyramid. Round to the nearest hundredth. b. Find the mass of the pyramid in grams. c. Science The density of a substance is the ratio of its mass to its volume. Find the density of the pyramid in grams per cubic centimeter. Round to the nearest hundredth. 322 Unit 5 © Houghton Mifflin Harcourt Publishing Company 20. Analyze Relationships There are two glass pyramids at the Louvre Museum in Paris, France. The outdoor pyramid has a square base with side lengths of 35.4 meters and a height of 21.6 meters. The indoor pyramid has a square base with side lengths of 15.5 meters and a height of 7 meters. How many times as great is the volume of the outdoor pyramid than that of the indoor pyramid? LESSON 10.2 ? Volume of Triangular Prisms and Pyramids ESSENTIAL QUESTION Equations, expressions, and relationships—7.9.A Solve problems involving the volume of rectangular prisms, triangular prisms, rectangular pyramids, and triangular pyramids. Also 7.8.B How do you find the volume of a triangular prism or a triangular pyramid? Finding the Volume of a Triangular Prism The volume V of a prism is the area of its base B times its height h, or V = Bh. Math On the Spot EXAMPL 1 EXAMPLE 7.9.A Find the volume of the triangular prism. STEP 1 5 ft 5 ft Find the area of the triangular base. 3 ft A = _12 bh Write the formula. A = _12 (8 · 3) Substitute 8 for the length of the triangular base and 3 for the height of the triangle. A = 12 ft2 © Houghton Mifflin Harcourt Publishing Company STEP 2 Write the formula. V = 12 · 7 Substitute the value from Step 1 for B and 7 for the height of the prism. 7 ft 8 ft Find the volume of the triangular prism. V = Bh my.hrw.com Math Talk Mathematical Processes Explain the difference between the variables b and B in the formulas A = _12 bh and V = Bh. V = 84 ft3 The volume of the triangular prism is 84 cubic feet. Reflect 1. What If? Suppose you know the volume V and base area B of a triangular prism. How could you find the height of the prism? YOUR TURN 2. Find the volume of a garden seat in the shape of a triangular prism with a height of 30 inches and a base area of 72 in2. Personal Math Trainer Online Assessment and Intervention my.hrw.com Lesson 10.2 323 EXPLORE ACTIVITY 7.8.B Exploring the Volume of a Triangular Pyramid Previously you explored the volumes of rectangular prisms and pyramids. Now you will repeat the same activity with triangular pyramids and prisms that have the same height and congruent bases. STEP 1 Make three-dimensional models. Make larger versions of the nets shown. Make sure the bases and heights in each net are the same size. Fold each net, and tape it together to form an open prism or pyramid. STEP 2 Fill the pyramid with beans. Make sure that the beans are level with the opening of the pyramid. How many pyramids full of beans do you think it will take to fill the prism? Pour the beans into the prism. Repeat until the prism is full. Was your conjecture supported? STEP 3 Write a fraction that compares the volume of the pyramid to the volume of the prism. 3. Analyze Relationships Does it appear that the relationship between the volume of triangular pyramids and prisms is the same as that for rectangular pyramids and prisms? 4. Draw Conclusions Write a formula for the volume of a triangular pyramid with a base area of B and a height of h. YOUR TURN Personal Math Trainer Online Assessment and Intervention my.hrw.com 324 Unit 5 5. The volume of a triangular pyramid is 13.5 m3. What is the volume of a triangular prism with a congruent base and the same height? Explain. © Houghton Mifflin Harcourt Publishing Company Reflect Solving Volume Problems As you solve volume problems, you will use the volume formulas you have learned. You will also need to use the formula for the area of a triangle: A = _12 bh. Math On the Spot EXAMPL 2 EXAMPLE 7.9.A Mr. Martinez is building wooden shapes for a sculpture in the park. His plans show a triangular pyramid and a triangular prism, and each shape is 5 feet high. The base of each shape is a triangle with a base of 2.5 feet and a height of 2 feet. How much greater than the volume of the pyramid is the volume of the prism? 5 ft The triangular base is the same for both shapes. Find the area of the base B. The area A of the triangle is 1 bh A = __ the same as B in V = Bh. 2 Substitute the values for b 1 (2.5)(2) = 2.5 A = __ 2 and h of the triangle. 5 ft STEP 1 my.hrw.com My Notes 2 ft 2.5 ft 2 ft 2.5 ft The area of the base B for both shapes is 2.5 ft2. STEP 2 Find the volume of the prism. V = Bh Use the formula. V = (2.5) × 5 = 12.5 Substitute the value for B and h of the prism. The volume of the prism is 12.5 ft3. STEP 3 Find the volume of the pyramid. © Houghton Mifflin Harcourt Publishing Company Volume of pyramid = _13 · volume of prism = _13 · 12.5 ≈ 4.2 Substitute and calculate. The volume of the pyramid is approximately 4.2 ft3. STEP 4 Compare the volumes. Volume of prism - volume of pyramid = 12.5 - 4.2 = 8.3 The volume of the prism is 8.3 ft3 greater than that of the pyramid. YOUR TURN 6. How much greater is the volume of a triangular prism with base area of 14 cm2 and height of 4.8 cm than the volume of a triangular pyramid with the same height and base area? Personal Math Trainer Online Assessment and Intervention my.hrw.com Lesson 10.2 325 Guided Practice 1. Find the volume of the triangular prism. (Example 1) Find the area of the base of the prism. Use the equation A = A= ( )( 16 in. bh. )= 3 in. in2 3 in. Find the volume of the prism. Use the equation V = Bh. V= ( )( )= in3 2. A triangular pyramid and a triangular prism have congruent bases and the same height. The triangular pyramid has a volume of 90 m3. Find the volume of the prism. (Explore Activity) The volume of the prism is is because the volume of the prism times the volume of the pyramid. 3. In Exercise 2, how much greater is the volume of the prism than the volume of the pyramid? (Example 2) Find the volume of each figure. 4. 5. 6. 7 in. 10 in. 6 yd 9 ft 5 in. 9 ft ? ? 6 yd 6 yd ESSENTIAL QUESTION CHECK-IN 7. A pyramid has a base that is a triangle. The length of the base of the triangle is 5 meters, and the height of the triangle is 12 meters. The height of the pyramid is 10 meters. How would you explain to a friend how to find the volume of the pyramid? 326 Unit 5 © Houghton Mifflin Harcourt Publishing Company 5 ft Name Class Date 10.2 Independent Practice Personal Math Trainer 7.9.A, 7.8.B my.hrw.com Online Assessment and Intervention 8. A trap for insects is in the shape of a triangular prism. The area of the base is 3.5 in2 and the height of the prism is 5 in. What is the volume of this trap? 9. Arletta built a cardboard ramp for her little brothers’ toy cars. Identify the shape of the ramp. Then find its volume. 6 in. 7 in. 25 in. 10. Represent Real-World Problems Sandy builds this shape of four congruent triangles using clay and toothpicks. The area of each triangle is 17.6 cm2, and the height of the shape is 5.2 cm. What three-dimensional figure does the shape Sandy built resemble? If this were a solid shape, what would be its volume? Round your answer to the nearest tenth. 11. Draw Conclusions Would tripling the height of a triangular prism triple its volume? Explain. 3.5 ft © Houghton Mifflin Harcourt Publishing Company 12. The Jacksons went camping in a state park. One of the tents they took is shown. What is the volume of the tent? 13. Shawntelle is solving a problem involving a triangular pyramid. You hear her say that “bee” is equal to 24 inches. How can you tell if she is talking about the base area B of the pyramid or about the base b of the triangle? 6 ft 4.5 ft 14. Alex made a sketch for a homemade soccer goal he plans to build. The goal will be in the shape of a triangular prism. The legs of the right triangles at the sides of his goal measure 4 ft and 8 ft, and the opening along the front is 24 ft. How much space is contained within this goal? Lesson 10.2 327 FOCUS ON HIGHER ORDER THINKING Work Area 15. A plastic puzzle in the shape of a triangular prism has bases that are equilateral triangles with side lengths of 4 inches and a height of 3.5 inches. The height of the prism is 5 inches. Find the volume of the prism. 16. Persevere in Problem Solving Lynette’s grandmother has a metal doorstop with the dimensions shown. Find the volume of the metal in the doorstop. The metal in the doorstop has a mass of about 8.6 grams per cubic centimeter. Find the mass of the doorstop. 10 cm 2.5 cm 8 cm 6 cm 17. Make a Conjecture Don and Kayla each draw a triangular pyramid that has a volume of 100 cm3. They do not draw identical shapes. Give a set of possible dimensions for each pyramid. 5 cm 19. Analyze Relationships What effect would doubling all the dimensions of a triangular pyramid have on the volume of the pyramid? Explain your reasoning. 328 Unit 5 © Houghton Mifflin Harcourt Publishing Company 18. Multistep Don’s favorite cheese snack comes in a box of six pieces. Each piece 4 cm of cheese has the shape of a triangular prism that is 2 cm high. The triangular base of the prism has a height of 5 cm 2 cm and a base of 4 cm. Find the volume of cheese in the box. LESSON 10.3 ? Lateral and Total Surface Area ESSENTIAL QUESTION Equations, expressions, and relationships— 7.9.D Solve problems involving the lateral and total surface area of a rectangular prism, rectangular pyramid, triangular prism, and triangular pyramid by determining the area of the shape’s net. How do you find the lateral and total surface area of rectangular and triangular prisms or pyramids? Lateral and Total Surface Area of a Prism The lateral faces of a prism are parallelograms that connect the bases. Each face that is not a base is a lateral face. The sum of the areas of all the lateral faces is the lateral area of the prism. The surface area is the sum of areas of all of the surfaces of a figure expressed in square units. The total surface area of a prism can be found by finding the sum of the lateral area and the area of the bases. Math On the Spot my.hrw.com Lateral face Base You can find the lateral area and total surface area of a prism by using a net. © Houghton Mifflin Harcourt Publishing Company EXAMPL 1 EXAMPLE 7.9.A A net of a rectangular prism is shown. Use the net to find the lateral area and the total surface area of the prism. Each square represents one square inch. The blue regions are the bases, and the green regions are the lateral faces. STEP 1 Find the lateral area of the rectangular prism. There are two 2 in. by 5 in. rectangles, and two 2 in. by 6 in. rectangles. 2 · (2 · 5) = 20 in2 2 · (2 · 6) = 24 in2. Math Talk Mathematical Processes Explain how the total surface area of a prism differs from the lateral area. The lateral area is 20 + 24 = 44 in2. STEP 2 Find the total surface area of the rectangular prism. Each base is 5 inches by 6 inches. 2 · (5 · 6) = 60 in2. The total surface area is 44 + 60 = 104 in2. Lesson 10.3 329 YOUR TURN Personal Math Trainer Online Assessment and Intervention 1. Use the net to find the lateral area and the total surface area of the triangular prism described by the net. 4 cm 3 cm 4 cm my.hrw.com 4 cm 5 cm Lateral and Total Surface Area of a Pyramid Math On the Spot my.hrw.com In a rectangular pyramid, the base is a rectangle, and the lateral faces are triangles. The lateral area and the total surface area are defined in the same way as they are for a prism. EXAMPLE 2 7.9.D Use the net of this rectangular pyramid to find the lateral area and the total surface area of this pyramid. 17 in. 16 in. 17 in. 16 in. 16 in. STEP 1 Find the lateral area of the rectangular pyramid. There are four triangles with base 16 in. and height 17 in. The lateral area is 4 × _12(16)(17) = 544 in2. STEP 2 Find the total surface area of the rectangular pyramid. The area of the base is 16 × 16 = 256 in2. The total surface area is 544 + 256 = 800 in2. Reflect Animated Math my.hrw.com 330 Unit 5 2. How many surfaces does a triangular pyramid have? What shape are they? © Houghton Mifflin Harcourt Publishing Company 16 in. YOUR TURN 2.6 ft 3 ft 3. The base and all three faces of a triangular pyramid are equilateral triangles with side lengths of 3 ft. The height of each triangle is 2.6 ft. Use the net to find the lateral area and the total surface area of the triangular pyramid. 4. Use a net to find the lateral area and the total surface area of the pyramid. Personal Math Trainer Online Assessment and Intervention my.hrw.com 20 in. 20 in. 16 in. 16 in. Solving Area Problems Several students have volunteered to design and build structures for the Lincoln Middle School art exhibit. EXAMPL 3 EXAMPLE 7.9.D © Houghton Mifflin Harcourt Publishing Company Shoshanna’s team plans to build stands to display sculptures. Each stand will be in the shape of a rectangular prism. The prism will have a square base with side lengths of 2 _12 feet, and it will be 3 _12 feet high. The team plans to cover the stands with metallic foil that costs $0.22 per square foot. How much money will the team save on each stand if they cover only the lateral area instead of the total surface area? STEP 1 Make a net of the rectangular prism. STEP 2 Find the lateral and total surface areas of the prism. 2.5 ft Math On the Spot my.hrw.com My Notes 2.5 ft 3.5 ft The four lateral faces are 3_12 feet by 2_12 feet rectangles. The two bases are 2_12 feet by 2_12 feet squares. ( ) Total surface area: 2 · (2_12 · 2_12) + 35 = 47_12 Lateral area: 4 · 3_12 · 2_12 = 35 square feet STEP 3 square feet Find and compare the prices. ( ) Cost for total surface area: $0.22 47_21 = $10.45 Add the area of the bases to the lateral area. Cost for lateral area: $0.22(35) = $7.70 The team will save $10.45 - $7.70, or $2.75, for each stand by covering only the lateral area. Lesson 10.3 331 YOUR TURN Personal Math Trainer Online Assessment and Intervention my.hrw.com 5. Kwame’s team will make two triangular pyramids to decorate the entrance to the exhibit. They will be wrapped in the same metallic foil. Each base is an equilateral triangle. If the base has an area of about 3.9 square feet, how much will the team save altogether by covering only the lateral area of the two pyramids? 6.1 ft 3 ft The foil costs $0.22 per square foot. Guided Practice A three-dimensional figure is shown sitting on a base. (Example 1) 1. The figure has a total of rectangular faces. 2. Of the total number of faces, 3. The figure is a 3 7 are lateral faces. 6 . 4. Sketch a net of the figure. 5. The lateral area of the prism is 6. The total surface area is . . A triangular prism is shown. (Example 2) 4.3 cm 6 cm 4.3 cm 4 cm 8. Find the lateral area of the prism. 9. Find the total surface area of the prism. 3 cm 10. Use a net to find the total surface area of the pyramid. Then find the cost of wrapping the pyramid completely in gold foil that costs $0.05 per square centimeter. (Examples 2 and 3) 12 cm 10 cm ? ? ESSENTIAL QUESTION CHECK-IN 11. How do you find the lateral and total surface area of a triangular pyramid? 332 Unit 5 10 cm © Houghton Mifflin Harcourt Publishing Company 7. Identify the number and type of faces of the prism. Name Class Date 10.3 Independent Practice Personal Math Trainer 7.9.D my.hrw.com Online Assessment and Intervention 12. Use a net to find the lateral area and the total surface area of the cereal box. Lateral area: 12 in. Total surface area: 13. Describe a net for the shipping carton shown. 8 in. 2 in. 3.6 in. 14. A shipping carton is in the shape of a triangular prism. Use a net to find the lateral area and the total surface area of the carton. Lateral area: 15 in. 3 in. Total surface area: 4 in. 15. Victor wrapped this gift box with adhesive paper (with no overlaps). How much paper did he use? 5 in. © Houghton Mifflin Harcourt Publishing Company 16. Vocabulary Name a three-dimensional shape that has four triangular faces and one rectangular face. 6 in. 8 in. 17. Cindi wants to cover the top and sides of this box with glass tiles that are 1 cm square. How many tiles will she need? 9 cm 20 cm 15 cm 18. A glass paperweight has the shape of a triangular prism. The bases are equilateral triangles with side lengths of 4 inches and heights of 3.5 inches. The height of the prism is 5 inches. Find the lateral area and the total surface area of the paperweight. 2.5 ft 2 ft 19. The doghouse shown has a floor, but no windows. Find the total surface area of the doghouse (including the door). 2.5 ft 2 ft 4 ft 20. Describe the simplest way to find the total surface area of a cube. 3 ft Lesson 10.3 333 21. Communicate Mathematical Ideas Describe how you approach a problem involving lateral area and total surface area. What do you do first? In what ways can you use the figure that is given with a problem? What are some shortcuts that you might use when you are calculating these areas? FOCUS ON HIGHER ORDER THINKING Work Area 22. Persevere in Problem Solving A pedestal in a craft store is in the shape of a triangular prism. The bases are right triangles with side lengths of 12 cm, 16 cm, and 20 cm. The store owner used 192 cm2 of burlap cloth to cover the lateral area of the pedestal. Find the height of the pedestal. 24. Critique Reasoning A triangular pyramid is made of 4 equilateral triangles. The sides of the triangles measure 5 m, and the height of each triangle is 4.3 m. A rectangular prism has a height of 4.3 m and a square base that is 5 m on each side. Susan says that the total surface area of the prism is more than twice the total surface area of the pyramid. Is she correct? Explain. 334 Unit 5 © Houghton Mifflin Harcourt Publishing Company 23. Communicate Mathematical Ideas The base of Prism A has an area of 80 ft2, and the base of Prism B has an area of 80 ft2. The height of Prism A is the same as the height of Prism B. Is the base of Prism A congruent to the base of Prism B? Explain. MODULE QUIZ Ready Personal Math Trainer 10.1 Volume of Rectangular Prisms and Pyramids Online Assessment and Intervention my.hrw.com Find the volume of each figure. 1. 2. 18 ft 16 ft 12 ft 10.2 Volume of Triangular Prisms and Pyramids Find the volume of each figure. 4. 3. 22 m 15 cm 17 cm 40 cm 5m 8 cm 15 m 10.3 Lateral and Total Surface Area Find the lateral and total surface area of each figure using its net. 5. 40 m 6. © Houghton Mifflin Harcourt Publishing Company 10 in. 15 m 24 m 19.2 m 19.2 m 6 in. 12 in. ESSENTIAL QUESTION 7. How can you use volume and surface area to solve real-world problems? Module 10 335 Personal Math Trainer MODULE 10 MIXED REVIEW Texas Test Prep Selected Response 1. The volume of a triangular pyramid is 232 cubic units. The area of the base of the pyramid is 29 square units. What is the height of the pyramid? A 8 units C B 12 units D 24 units my.hrw.com 6. What is the total surface area of the square pyramid whose net is shown? 10 in. 16 in. 16 units 2. What is the volume of a rectangular prism that has a length of 8.5 centimeters (cm), a width of 3.2 centimeters, and a height of 6 centimeters? A 256 in2 B 336 in2 C A 19.2 cm3 C B 27.2 cm3 D 163.2 cm3 51 cm3 3. What is the volume of the rectangular pyramid shown? 576 in2 D 896 in2 Gridded Response 7. What is the lateral area in square meters of the prism? 22 yd 20 yd 10 m 8m 15 yd 12 m A 1,650 yd C B 2,200 yd3 D 6,600 yd3 3,300 yd 3 4. A circle has a circumference of 56π centimeters (cm). What is the radius of the circle? A 28 cm C B 56 cm D 112 cm 88 cm 5. What is the volume of a triangular prism that has a height of 45 meters and has a base with an area of 20 square meters? A 225 m3 C B 300 m3 D 900 m3 Unit 5 450 m3 18 m . 0 0 0 0 0 0 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 7 7 7 7 7 7 8 8 8 8 8 8 9 9 9 9 9 9 © Houghton Mifflin Harcourt Publishing Company 3 336 Online Assessment and Intervention UNIT 5 Study Guide MODULE ? 9 Review Applications of Geometry Concepts ESSENTIAL QUESTION How can you apply geometry concepts to solve real-world problems? EXAMPLE 1 Find (a) the value of x and (b) the measure of ∠APY. a. ∠XPB and ∠YPB are supplementary. A adjacent angles (ángulos adyacentes) complementary angles (ángulos complementarios) congruent angles (ángulos congruentes) supplementary angles (ángulos suplementarios) vertical angles (ángulos opuestos por el vértice) 3x + 78° = 180° Y P 3x 78° X Key Vocabulary 3x = 102° B x = 34° b. ∠APY and ∠XPB are vertical angles. m∠APY = m∠XPB = 3x = 102° EXAMPLE 2 Find the area of the composite figure. It consists of a semicircle and a rectangle. Area of semicircle = 0.5(πr2) ≈ 0.5(3.14)25 10 cm © Houghton Mifflin Harcourt Publishing Company ≈ 39.25 cm2 Area of rectangle = ℓw 6 cm = 10(6) = 60 cm2 The area of the composite figure is approximately 99.25 square centimeters. EXERCISES 1. Find the value of y and the measure of ∠YPS (Lesson 9.1) R 140° 5y P Y y= Z m ∠YPS = S Unit 5 337 Find the circumference and area of each circle. Round to the nearest hundredth. (Lessons 9.2, 9.3) 2. 3. 4.5 m 22 in. Find the area of each composite figure. Round to the nearest hundredth if necessary. (Lesson 9.4) 4. 5. 9 in. 9 in. 20 cm 13 in. 16 cm Area MODULE MODULE ? Area 10 10 Volume and Surface Area Key Vocabulary ESSENTIAL QUESTION How can you use volume and surface area to solve real-world problems? EXAMPLE The height of the figure whose net is shown is 8 feet. Identify the figure. Then find its volume, lateral area, and total surface area. lateral area (área lateral) lateral faces (cara lateral) net (plantilla) pyramid (pirámide) total surface area (área de superficie total) The figure is a rectangular pyramid. = _13 (144)(8) Lateral Area () 4 _12 (12)(8) = 192 Total Surface Area 8 ft 12 ft 192 + 122 = 192 + 144 = 384 The volume of the rectangular pyramid is 384 cubic feet, the lateral surface area is 192 square feet, and the total surface area is 336 square feet. = 336 © Houghton Mifflin Harcourt Publishing Company Volume = _13 Bh EXERCISES 1. Identify the figure represented by the net. Then find its lateral area and total surface area. (Lesson 10.3) 10 ft 15 ft 13 ft 13 ft 13 ft 338 2. Unit 5 13 ft Find the volume of each figure. (Lessons 10.1, 10.2) 2. 3. 9m 12 in. 6m 5 in. 7 in. 4m 4. The volume of a rectangular pyramid is 1.32 cubic inches. The height of the pyramid is 1.1 inches and the length of the base is 1.2 inches. Find the width of the base of the pyramid. (Lesson 10.1) 5. The volume of a triangular prism is 264 cubic feet. The area of a base of the prism is 48 square feet. Find the height of the prism. (Lesson 10.2) Unit 5 Performance Tasks 1. CAREERS IN MATH Product Design Engineer Miranda is a product design engineer working for a sporting goods company. She designs a tent in the shape of a triangular prism. The dimensions of the tent are shown in the diagram. 6 ft 1 9— 2 ft 8 ft © Houghton Mifflin Harcourt Publishing Company a. Draw a net of the triangular prism and label the dimensions. 1 7— 4 ft Unit 5 339 Unit 5 Performance Tasks (cont'd) b. How many square feet of material does Miranda need to make the tent (including the floor)? Show your work. c. What is the volume of the tent? Show your work. d. Suppose Miranda wants to increase the volume of the tent by 10%. The specifications for the height (6 feet) and the width (8 feet) must stay the same. How can Miranda meet this new requirement? Explain. 2. Li is making a stand to display a sculpture made in art class. The stand will be 45 centimeters wide, 25 centimeters long, and 1.2 meters high. a. What is the volume of the stand? Write your answer in cubic centimeters. b. Li needs to fill the stand with sand so that it is heavy and stable. Each piece of wood is 1 centimeter thick. The boards are put together as shown in the figure, which is not drawn to scale. How many cubic centimeters 1.2 m of sand does she need to fill the stand? Explain how you found your answer. 1 cm 1.2 m 25 cm Side View © Houghton Mifflin Harcourt Publishing Company 45 cm Front View 340 Unit 5 Personal Math Trainer UNIT 5 MIXED REVIEW Texas Test Prep Selected Response 1. The dimensions of the pyramid are given in centimeters (cm). What is the volume of the pyramid? B 1.25x + 15 = 20 10 cm C 15x - 1.25 = 20 9 cm D 1.25x - 15 = 20 A 222.5 cubic centimeters 5. A bank offers a home improvement loan with simple interest at an annual rate of 12%. J.T. borrows $14,000 over a period of 3 years. How much will he pay back altogether? B 330 cubic centimeters C 445 cubic centimeters D 990 cubic centimeters 2. The volume of a triangular pyramid is 437 cubic units. The height of the pyramid is 23 units. What is the area of the base of the pyramid? © Houghton Mifflin Harcourt Publishing Company 4. A one-topping pizza costs $15.00. Each additional topping costs $1.25. Let x be the number of additional toppings. You have $20 to spend. Which equation can you solve to find the number of additional toppings you can get on your pizza? A 15x + 1.25 = 20 11 cm A 6.3 square units C 38 square units B 19 square units D 57 square units Hot ! Tip my.hrw.com Online Assessment and Intervention Make sure you look at all the answer choices before making your decision. Try substituting each answer choice into the problem if you are unsure of the answer. 3. What is the lateral surface area of the square pyramid whose net is shown? 9 ft A $15,680 B $17,360 C $19,040 D $20,720 6. What is the volume of a triangular prism that is 75 centimeters long and that has a base with an area of 30 square centimeters? A 2.5 cubic centimeters B 750 cubic centimeters C 1,125 cubic centimeters D 2,250 cubic centimeters 12 ft A 216 square feet B 388 square feet C 568 square feet D 776 square feet Unit 5 341 7. The radius of the circle is given in meters. What is the circumference of the circle? Use 3.14 for π. 10. What is the volume in cubic centimeters of a rectangular prism that has a length of 6.2 centimeters, a width of 3.5 centimeters, and a height of 10 centimeters? 8m . 0 0 0 0 0 0 1 1 1 1 1 1 2 2 2 2 2 2 B 50.24 meters 3 3 3 3 3 3 C 200.96 meters 4 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 7 7 7 7 7 7 8 8 8 8 8 8 9 9 9 9 9 9 A 25.12 meters D 803.84 meters 8. The dimensions of the figure are given in millimeters. What is the area of the two-dimensional figure? 13 mm Hot ! Tip 6 mm 13 mm A 39 square millimeters It is helpful to draw or redraw a figure. Answers to geometry problems may become clearer as you redraw the figure. 11. What is the area of the circle in square meters? Use 3.14 for π. B 169 square millimeters C 208 square millimeters 18 m D 247 square millimeters Gridded Response 9. What is the measure in degrees of an angle that is supplementary to a 74° angle? . 342 Unit 5 0 0 0 0 0 0 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 0 0 0 0 0 0 4 4 4 4 4 4 1 1 1 1 1 1 5 5 5 5 5 5 2 2 2 2 2 2 6 6 6 6 6 6 3 3 3 3 3 3 7 7 7 7 7 7 4 4 4 4 4 4 8 8 8 8 8 8 5 5 5 5 5 5 9 9 9 9 9 9 6 6 6 6 6 6 7 7 7 7 7 7 8 8 8 8 8 8 9 9 9 9 9 9 © Houghton Mifflin Harcourt Publishing Company .